The Vulnerability of Some Networks including Cycles via Domination Parameters

Tufan Turaci,Hüseyin Aksan

doi:10.1155/2016/4705493

Abstract

LetG=(V(G),E(G))be an undirected simple connected graph. A network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. Finding the vulnerability values of communication networks modeled by graphs is important for network designers. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. The domination number and its variations are the most important vulnerability parameters for network vulnerability. Some variations of domination numbers are the 2-domination number, the bondage number, the reinforcement number, the average lower domination number, the average lower 2-domination number, and so forth. In this paper, we study the vulnerability of cycles and related graphs, namely, fans,k-pyramids, andn-gon books, via domination parameters. Then, exact solutions of the domination parameters are obtained for the above-mentioned graphs.

Highlights

Graph theory has become one of the most powerful mathematical tools in the analysis and study of the architecture of a network
Various measures were defined to measure the robustness of network and a variety of graph theoretic parameters have been used to derive formulas to calculate network vulnerability
Graph vulnerability relates to the study of graph when some of its elements are removed

Summary

Introduction

Graph theory has become one of the most powerful mathematical tools in the analysis and study of the architecture of a network. Various measures were defined to measure the robustness of network and a variety of graph theoretic parameters have been used to derive formulas to calculate network vulnerability. Graph vulnerability relates to the study of graph when some of its elements (vertices or edges) are removed. The vertex (edge) connectivity is defined to be the minimum number of vertices (edges) whose deletion results in a disconnected or trivial graph [7]. A natural way to model the topology of a communications network is as a graph consisting of vertices and edges. Wheels, pyramids, bipyramids, and ncycle books are among such graphs

The Average Lower Domination Number

Calculation of ALDN and AL2DN of Cycles and Related Networks

Conclusion